Vol. 8, No. 8, 2014

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ISSN: 1944-7833 (e-only)
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Proper triangular $\mathbb{G}_{a}$-actions on $\mathbb{A}^{4}$ are translations

Adrien Dubouloz, David R. Finston and Imad Jaradat

Vol. 8 (2014), No. 8, 1959–1984
Abstract

We describe the structure of geometric quotients for proper locally triangulable Ga-actions on locally trivial A3-bundles over a nœtherian normal base scheme X defined over a field of characteristic 0. In the case where dimX = 1, we show in particular that every such action is a translation with geometric quotient isomorphic to the total space of a vector bundle of rank 2 over X. As a consequence, every proper triangulable Ga-action on the affine four space Ak4 over a field of characteristic 0 is a translation with geometric quotient isomorphic to Ak3.

Keywords
proper additive group actions, geometric quotients, principal homogeneous bundles, affine fibrations
Mathematical Subject Classification 2010
Primary: 14L30
Secondary: 14R10, 14R20, 14R25
Milestones
Received: 23 April 2014
Accepted: 10 September 2014
Published: 28 November 2014
Authors
Adrien Dubouloz
CNRS, Institut de Mathématiques de Bourgogne
Université de Bourgogne
9 Avenue Alain Savary
BP 47870
21078 Dijon
France
David R. Finston
Department of Mathematical Sciences
New Mexico State University
Las Cruces, NM 88003
United States
Imad Jaradat
Department of Mathematical Sciences
Jordan University of Science and Technology
P.O.Box 3030
Irbid 22110
Jordan